You don't call Customer Care. You call the Partner Desk.
http://www.alaskaair.com/content/abo...tact-info.aspx
"Partner Award Reservations: 1-800-307-6912
5:00 a.m. - Midnight (PT), Daily"
BA and AF are not searchable from alaskaair.com. As a matter of fact, it says so here:
http://www.alaskaair.com/content/mil...w.aspx#partner
"Partner award travel with American Airlines and Delta Air Lines can be booked online. All other Partner award travel may be booked through the Partner Desk at 1-800-307-6912, daily between 5:00 a.m. and midnight (PT)."